The high power modules are notoriously sensible to power instability,
and don't like noise on the power supply.
You could swap out components to find the culprit.
Does it work with one high power module? On which side?
Two AA batteries make a good stable NRF power supply, for tests.
But good to hear that it works. 
Oh I see! Seems they are a little "special" and not so easy to handle. The PCB antenna type seems to be more forgiving. 
In the end I do need the antenna version as the range is too long to get to the basement where the mailbox is located so I have to experiment how to get the antenna version to run reliably. Its a 5 floors building but with a quite wide stairway so I hope the antenna version will get down there. The PCB antenna version gets to the middle at least. 
Thanks!
Anyhow: thanks for your time and help!